Determining the projections onto the Horizontal and Vertical Space I am looking at a map $\Pi: GL(n)\mapsto Sym^+(n)$ via $\Pi(A)=AA^T$.
This has differential $d\Pi_A(Z)=ZA^T+AZ^T$. 
The vertical space $\mathcal{V}_A$ is the kernel of $d\Pi_A$, and so
$$Z\in\mathcal{V_A}\iff Z^T=-A^{-1}ZA^T.$$
With the inner product $G(Z,W)=tr(Z^TW)$, the horizontal space $\mathcal{H}_A$ is the orthogonal complement to the Vertical space. 
We have a defining property for $\mathcal{H}_A$ which is 
$$W\in\mathcal{H}_A\iff Z^T=A^TZA^{-1}.$$
I am able to prove, this but not in a satisfying way (Go right to left and then by dimensions being equal they must define the same sets), so seeing how this relationship is derived would be helpful. 
My big hang up is that I can't figure out what the projections onto these subspaces look like. Thanks. 
 A: Note that $Z\in\mathcal V_A \iff AZ^\top$ is skew-symmetric. With respect to the standard inner product you've given on $n\times n$ matrices, the orthogonal complement of the subspace of skew-symmetric matrices is the subspace of symmetric matrices.
Since $\mathcal V_A$ is the set of all matrices $Z$ so that $AZ^\top$ is skew-symmetric, we see that $\mathcal H_A$ must be the set of all matrices $W$ so that $WA^{-1}$ is symmetric. We know that $\langle AZ^\top,S\rangle = 0$ for all symmetric matrices $S$, and so $0 = \langle Z^\top,A^\top S \rangle = \langle Z,SA\rangle$. Thus $(\mathcal V_A)^\perp = \{W: WA^{-1} \text{ is symmetric}\}$. 
You can easily give a (sadly, non-orthogonal) basis for $\mathcal H_A$ by multiplying the obvious orthonormal basis for the subspace of symmetric  matrices on the right by $A$. Similarly, you obtain a basis for $\mathcal V_A$ by multiplying the obvious orthonormal basis for the subspace of skew-symmetric matrices on the right by $(A^\top)^{-1}$.
Given $X$ and $A$, to find the decomposition $X=Z+W$ with $Z\in\mathcal V_A$ and $W\in\mathcal H_A$, we can solve the equations
\begin{align*}
XA^\top + AX^\top &= WA^\top + AW^\top \\
W^\top &= A^\top W A^{-1}
\end{align*}
as a monster system of linear equations for $W$:
$$WA^\top + AA^\top W A^{-1} = AX^\top + XA^\top.$$
